Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family F(a,x) of maps of the interval [0,1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps analytically, where the results thus obtained have been approved with numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain parameter values, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameter whose Lyapunov characteristic exponent begins to be positive.Comment: 18 pages (Latex), 7 figure

Etale covers of affine spaces in positive characteristic

We prove that every projective variety of dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and a chosen smooth point not on the divisor to some point not in H.Comment: 8 pages; uses AMSLaTe